Abstract
We consider a nonlinear integral equation which can be interpreted as a generalization of Theodorsen’s nonlinear integral equation. This equation arises in computing the conformal mapping between simply connected regions. We present a numerical method for solving the integral equation and prove the uniform convergence of the numerical solution to the exact solution. Numerical results are given for illustration.
1. Introduction
Numerical methods for conformal mapping from a simply connected region onto another simply connected region are available only when one of the region is a standard region, mostly the unit disk . Let and be bounded simply connected regions in the -plane and -plane, respectively, such that their boundaries and are smooth Jordan curves. Then the mapping is calculated as the composition of the maps .
Recently, a numerical method has been proposed in [1] for direct approximation of the mapping . Assume that and are star-like with respect to the origin and defined by polar coordinates respectively, such that both and are -periodic continuously differentiable positive real functions with nonvanishing derivatives. By the Riemann-mapping theorem, there exists a unique conformal mapping function normalized by , . The boundary value of the function is on the boundary and can be described as where is the boundary correspondence function of the mapping function . The function is a strictly increasing function so that is a -periodic function.
The function is the unique solution of a nonlinear integral equation which can be interpreted as a generalization of Theodorsen’s nonlinear integral equation [1]. The proof of the existence and the uniqueness of the solution of the nonlinear integral equation was given in [1] for regions of which boundaries satisfy the so-called -condition; that is,
In this paper, the nonlinear integral equation is solved by an iterative method. Each iteration of the iterative method requires solving an linear system which is obtained by discretizing the integrals in the integral equation by the trapezoidal rule. The linear system is solved by a combination of the generalized minimal residual (GMRES) method and the fast multipole method (FMM) in operations. The main objective of this paper is to prove the uniform convergence of the numerical solution to the exact solution. We also study the properties of the generalized conjugation operator. Numerical results are presented for illustration.
2. Auxiliary Materials
2.1. The Functions and
Let be the mapping function from the simply connected region onto the unit disk with the normalization and . Then the boundary value of the function is on the unit circle and can be described as The function is the boundary correspondence function of the mapping function where is a -periodic function and for all . Let be the inverse of the function . Then is the boundary correspondence function of the inverse mapping function from onto ; that is, where is a -periodic function and for all .
2.2. The Norms
Let be the space of all real Hölder continuous -periodic functions on . With the inner product the space is a pre-Hilbert space. We define the norm by
Since and if , we have With the norm , we define a norm by We define also the maximum norm by Since , we have which implies that
Theorem 1. If , then
Proof. Since and if , we have Thus, it follows from [2, page 68] that We have also Hence, Since is bijective, we have Hence, (15) and (18) imply that Then (13) follows from (9), (17), and (19).
2.3. The Operators K and J
The conjugation operator is defined by Let be the operator defined by Hence, the operators and satisfy [3]
3. The Generalized Conjugation Operator
Let be the complex -periodic continuously differentiable function: We define the real kernels and as real and imaginary parts: The kernel is called the generalized Neumann kernel formed with and . The kernel is continuous and the kernel has the representation with a continuous kernel . See [4] for more details.
We define the Fredholm integral operators and and the singular integral operator on by We define an operator on by The operator is singular but bounded on [1]. Finally, we define an operator by
Remark 2. When reduces to the unit, then , the operator reduces to the operator , and the operator reduces to the operator ; that is, the operator is a generalization of the well-known conjugation operator (see [1] for more details).
The operator is related to the operator by [1] Since and , it follows from (29) that
Lemma 3 (see [1]). Let be given functions. Then is the boundary value of an analytic function in with if and only if
Lemma 4 (see [1]). If and , then with a real constant where is the unique analytic function in with the boundary values and .
Lemma 5 (see [1]). The operator has the following properties:
Lemma 6. The operator has the norm
Proof. The operator has the norm [1]. Since , hence . Hence, we obtain (33).
Lemma 7. The operators and satisfy
Proof. For any , let , , and . Then, it follows from (29) that . Thus, which by (22) implies that Since (36) holds for all functions , the operator identity (34) follows.
Lemma 8. The operator satisfies
Proof. Let and . Then, by Lemma 4, with a real constant . By the definition of the operator , we have . Since , we have Hence, holds for all . Thus, the operator identities (37) follow.
Lemma 9. For all functions , we have with equality for all with .
Proof. For all functions , the inequality (40) follows from (33).
For all functions with , we have from (37) that . Hence
which means that .
Theorem 10. Let and . Then
Proof. For and , we have from (29) that . Then, it follows from [2, page 64] that Hence, by (30), we have which implies that Hence, we obtain (42).
4. The Generalized Theodorsen Nonlinear Integral Equation
The boundary correspondence function is the unique solution of the nonlinear integral equation which is a generalization of the well-known Theodorsen integral equation [1]. Nonlinear integral equation (46) can be solved by the iterative method Then we have [1] Thus, if the curve satisfies the -condition (3), then That is, the approximate solutions converge to with respect to the norm if .
In this section, we will prove the uniform convergence of the approximate solutions to the exact solution . We will use the approach used in the proof of Proposition 1.5 in [2, page 69] related to Theodorsen’s integral equation. See also [5, 6].
Lemma 11. Consider
Proof. The function is the boundary correspondence function of the conformal mapping from onto the unit disk. Hence, the function satisfies [1] Then (50) follows from (51).
The previous lemma implies that (46) can be rewritten as and (47) can be rewritten as Thus
Lemma 12. Consider
Proof. Let be such that Then Hence, Thus Since and , we have which implies that Hence (55) follows from (48).
Lemma 13. Consider
Proof. We have
Since
, and , we obtain
Similarly, we have
In view of Theorem 10, it follows from (52) and (53) that
Hence, it follows from (68) that
By (70) and (66), we have
Hence,
Similarly, it follows from (69) that
which by (67) implies that
Hence,
which, in view of (67), implies that
Then (63) follows from (64), (72), and (76).
Theorem 14. If , then the approximate solution converges uniformly to the exact solution with
Proof. In view of (54), Lemma 7 implies that Thus, we have from (13) that Hence (77) follows from (55) and (63).
The following corollary follows from the previous theorem.
Corollary 15. If then
Remark 16. When reduces to the unit, then Hence, the results presented in this section reduces to the results presented in [2] for Theodorsen’s integral equation.
5. Discretizing (47)
In this paper, we will discretize (47) instead of (46). The numerical method used here is based on strict discretization of the integrals in the operator by the trapezoidal rule which gives accurate results since the integrals are over -periodic. Let be a given even positive integer. We define equidistant collocation points in the interval by Then, for -periodic function , the trapezoidal rule approximates the integral by If the function is continuous, then . If the integrand is times continuously differentiable, then the rate of convergence of the trapezoidal rule is . For analytic , the rate of convergence is better than for any positive integer [7, page 83]. See also [8].
For , the integral operator will be discretized by the Nyström method as follows: Hence, we have Since the kernel is continuous on both variables and since the function is continuous, we have [9]
The integral operator will be discretized by the Nyström method as follows: Since the kernel is continuous on both variables and since the function is continuous, we have [9]
To discretize the operator , we first approximate the function by the interpolating trigonometric polynomial of degree which interpolates at the points , . That is, Then is approximated by where [6]
The integral operator is then discretized by Then, it follows from (90) and (93) that The operator is bounded operator since the operator is bounded ( is continuous) and the operator is bounded operator (see [6]).
Since the kernel is continuous and is not an eigenvalue of the kernel [1], the operators are invertible and are uniformly bounded for sufficiently large [9]. Hence, we discretize the operator by the bounded operator
Lemma 17. If , then
Proof. Let and , then Let also be the unique solution of the discretized equation Thus, we have Since the kernel is continuous and is the discretization of , then it follows from [9, page 108] that Since is bounded and is continuous, then (95) implies that which with (100) and (101) implies (97).
To calculate the function in (47) for a given , we replace the operator in (47) by the approximate operator to obtain where is an approximation to . Substituting and , in (103) we obtain Equation (104) can be rewritten as which represents an linear system for the unknown . By obtaining for , the function can be calculated for by the Nyström interpolating formula. In the following lemma, we prove the uniform convergence of the approximate solution of discretized equation (103) to the solution of (46).
Lemma 18. Consider
Proof. Let . Then, we have Hence, The lemma is then followed from (97).
The proof of the uniform convergence of the approximate solution to the boundary correspondence function is given in the following theorem.
Theorem 19. If , then
Proof. We have Since , it follows from (77) that as . The theorem is then followed from (106).
6. The Algebraic System
Let be the vector where denotes transposition. Then, for any function defined on , we define as the vector obtained by componentwise evaluation of the function at the points , . As in MATLAB, for any two vectors and , we define as the componentwise vector product of and . If , for all , we define as the componentwise vector division of by . For simplicity, we denote by and by .
Let (given) and (unknown). Then system (105) can be rewritten as where is the identity matrix, is the discretized matrix of the operator , and is the discretized matrix of the operator [1]. Linear system (111) is uniquely solvable [4, 10, 11].
We start the iteration in (47) with and iterate until where tol is a given tolerance; that is, we start the iteration in (111) with and iterate until . Each iteration in (111) requires solving a linear system for given . Linear system (111) is solved in operations by the fast method presented in [11, 12] which is based on a combination of the MATLAB function gmres and the MATLAB function zfmm2dpart in the MATLAB toolbox FMMLIB2D [13]. In the numerical results below, for function zfmm2dpart, we assume that iprec which means that the tolerance of the FMM is . For the function gmres, we choose the parameters restart , gmrestol , and maxit , which means that the GMRES method is restarted every inner iterations, the tolerance of the GMRES method is , and the maximum number of outer iterations of GMRES method is . See [11, 12] for more details.
By obtaining , we obtain the values for . Then, the function can be calculated for by the Nyström interpolating formula. The convergence of to follows from Theorem 19. Then the values of the mapping function can be computed from (2). The interior values of the mapping function can be computed by the Cauchy integral formula which can be computed using the fast method presented in [12].
7. Numerical Examples
In this section, we will compute the conformal mapping from three simply connected regions , , and onto three simply connected regions , , and . The boundaries , , and of the regions , , and are parameterized by where the function is given by The boundaries , , and of the regions , , and are parameterized by where the function is given by The curves , , and satisfy the -condition where The numerical results obtained with and are shown in Figures 1, 2, and 3. The error norm versus the iteration number in (111) is shown in Figure 4. It is clear from Figure 4 that the number of iterations in (111) depends only on the boundary of the image region. More precisely, it depends on . The iterations in (111) converge only if . For small , a few number of iterations are required for convergence. For values of close to , a large number of iterations are required for convergence.
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Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.